The Random Projection Method for Stiff Multi-species Detonation Computation

Abstract

In this note we review the random projection method, recently introduced by the authors, for numerical simulations of the hyperbolic conservation laws with stiff reaction terms: $$ {U_t} + F{(U)_x} = \frac{1}{\varepsilon }\Psi (U) $$ . In this problem, the reaction timeEis small, making the problem numerically stiff. A classic spurious numerical phenomenon - the incorrect shock speed - occurs when the reaction time scale is not properly resolved numerically. The random projection method, a fractional step method that solves the homogeneous convection by any shock capturing method, followed by a random projection for the reaction term, was introduced in [1] to handle this numerical difficulty. For a scalar model problem, one can prove that the random projection methods capture the correct shock speed with a first order accuracy, if a monotonicity-preserving method is used in the convection step. This method can be extended to compute stiff multi-species detonation waves by randomizing the ignition temperatures. Numerical results of a problem with five species and two reactions will be presented.